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Nothing special, according to Plato. This comes from the end of book 10 of the Republic (X.621):

σκηνᾶσθαι οὖν σφᾶς ἤδη ἑσπέρας γιγνομένης παρὰ τὸν Ἀμέλητα ποταμόν, οὗ τὸ ὕδωρ ἀγγεῖον οὐδὲν στέγειν.
Then, as dusk was falling, they set up camp near the river Amelēta ("Not-Caring"), the waters of which couldn't be kept in any container.

μέτρον μὲν οὖν τι τοῦ ὕδατος πᾶσιν ἀναγκαῖον εἶναι πιεῖν, τοὺς δὲ φρονήσει μὴ σῳζομένους πλέον πίνειν τοῦ μέτρου· τὸν δὲ ἀεὶ πιόντα πάντων ἐπιλανθάνεσθαι.
Then every one of them needed to drink a specific amount of the water, though the ones who weren't saved by their good judgement drank more than that amount: and each one of them, at the moment of drinking, forgot everything (or "had everything wiped away").

ἐπειδὴ δὲ κοιμηθῆναι καὶ μέσας νύκτας γενέσθαι, βροντήν τε καὶ σεισμὸν γενέσθαι, καὶ ἐντεῦθεν ἐξαπίνης ἄλλον ἄλλῃ φέρεσθαι ἄνω εἰς τὴν γένεσιν, ᾄττοντας ὥσπερ ἀστέρας.
After they'd gone to bed, and midnight had come, there was thunder and an earthquake, and all of a sudden they were borne up away from that place to their birth, each one in a different way, like meteors.

αὐτὸς δὲ τοῦ μὲν ὕδατος κωλυθῆναι πιεῖν· ὅπῃ μέντοι καὶ ὅπως εἰς τὸ σῶμα ἀφίκοιτο, οὐκ εἰδέναι, ἀλλ’ ἐξαίφνης ἀναβλέψας ἰδεῖν ἤδη ἕωθεν αὑτὸν κείμενον ἐπὶ τῇ πυρᾷ.
But he (Ēros) was prevented from drinking the water: indeed, how and in what way he came back to his body, he had no idea, but all of a sudden he looked up to see that it was dawn, and he was lying on his funeral pyre.

(Translation mine.)

Plato never mentions anything in particular happening to the ones who drank too much; after all, drinking any of the water seems to erase your entire memory.

It's worth noting that the Orphics (who to some extent influenced Socrates and Plato) considered drinking from Lethë at all to be a failure: their initiates were taught to avoid Lethë entirely and drink from the pool of Mnemosynë ("memory") instead. The goal was, eventually, to break free of the cycle of reincarnation.